Minimal model of self-organized clusters with phase transitions in ecological communities

In complex ecological communities, species may self-organize into clusters or clumps where highly similar species can coexist. The emergence of such species clusters can be captured by the interplay between neutral and niche theories. Based on the generalized Lotka-Volterra model of competition, we propose a minimal model for ecological communities in which the steady states contain self-organized clusters. In this model, species compete only with their neighbors in niche space through a common interaction strength. Unlike many previous theories, this model does not rely on random heterogeneity in interactions. Even in this minimal model where only the common interaction strength is varied, we find an exponentially large set of states that exhibit a rich variety of cluster patterns with different sizes and combinations. There are sharp phase transitions into the formation of clusters. There are also multiple phase transitions between different sets of possible cluster patterns, many of which accumulate near a small number of critical points. We analyze this phase structure using both numerical and analytical methods. In addition, the special case with only nearest-neighbor interactions is exactly solvable using the method of transfer matrices from statistical mechanics. We analyze the critical behavior of these systems.